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Theorem itg1addlem4 25063
Description: Lemma for itg1add 25066. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
itg1add.4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
Assertion
Ref Expression
itg1addlem4 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   𝜑,𝑖,𝑗,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)

Proof of Theorem itg1addlem4
Dummy variables 𝑤 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5 (𝜑𝐹 ∈ dom ∫1)
2 i1fadd.2 . . . . 5 (𝜑𝐺 ∈ dom ∫1)
31, 2i1fadd 25059 . . . 4 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
4 itg1add.4 . . . . . . 7 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
5 ax-addf 11130 . . . . . . . . 9 + :(ℂ × ℂ)⟶ℂ
6 ffn 6668 . . . . . . . . 9 ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
75, 6ax-mp 5 . . . . . . . 8 + Fn (ℂ × ℂ)
8 i1frn 25041 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
91, 8syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 ∈ Fin)
10 i1frn 25041 . . . . . . . . . 10 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
112, 10syl 17 . . . . . . . . 9 (𝜑 → ran 𝐺 ∈ Fin)
12 xpfi 9261 . . . . . . . . 9 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
139, 11, 12syl2anc 584 . . . . . . . 8 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
14 resfnfinfin 9276 . . . . . . . 8 (( + Fn (ℂ × ℂ) ∧ (ran 𝐹 × ran 𝐺) ∈ Fin) → ( + ↾ (ran 𝐹 × ran 𝐺)) ∈ Fin)
157, 13, 14sylancr 587 . . . . . . 7 (𝜑 → ( + ↾ (ran 𝐹 × ran 𝐺)) ∈ Fin)
164, 15eqeltrid 2842 . . . . . 6 (𝜑𝑃 ∈ Fin)
17 rnfi 9279 . . . . . 6 (𝑃 ∈ Fin → ran 𝑃 ∈ Fin)
1816, 17syl 17 . . . . 5 (𝜑 → ran 𝑃 ∈ Fin)
19 difss 4091 . . . . 5 (ran 𝑃 ∖ {0}) ⊆ ran 𝑃
20 ssfi 9117 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 ∖ {0}) ⊆ ran 𝑃) → (ran 𝑃 ∖ {0}) ∈ Fin)
2118, 19, 20sylancl 586 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ∈ Fin)
22 ffun 6671 . . . . . . . . . . 11 ( + :(ℂ × ℂ)⟶ℂ → Fun + )
235, 22ax-mp 5 . . . . . . . . . 10 Fun +
24 i1ff 25040 . . . . . . . . . . . . . . 15 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
251, 24syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:ℝ⟶ℝ)
2625frnd 6676 . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ ℝ)
27 ax-resscn 11108 . . . . . . . . . . . . 13 ℝ ⊆ ℂ
2826, 27sstrdi 3956 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 ⊆ ℂ)
29 i1ff 25040 . . . . . . . . . . . . . . 15 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
302, 29syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺:ℝ⟶ℝ)
3130frnd 6676 . . . . . . . . . . . . 13 (𝜑 → ran 𝐺 ⊆ ℝ)
3231, 27sstrdi 3956 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 ⊆ ℂ)
33 xpss12 5648 . . . . . . . . . . . 12 ((ran 𝐹 ⊆ ℂ ∧ ran 𝐺 ⊆ ℂ) → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
3428, 32, 33syl2anc 584 . . . . . . . . . . 11 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (ℂ × ℂ))
355fdmi 6680 . . . . . . . . . . 11 dom + = (ℂ × ℂ)
3634, 35sseqtrrdi 3995 . . . . . . . . . 10 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ dom + )
37 funfvima2 7181 . . . . . . . . . 10 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
3823, 36, 37sylancr 587 . . . . . . . . 9 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
39 opelxpi 5670 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ⟨𝑥, 𝑦⟩ ∈ (ran 𝐹 × ran 𝐺))
4038, 39impel 506 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → ( + ‘⟨𝑥, 𝑦⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
41 df-ov 7360 . . . . . . . 8 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
424rneqi 5892 . . . . . . . . 9 ran 𝑃 = ran ( + ↾ (ran 𝐹 × ran 𝐺))
43 df-ima 5646 . . . . . . . . 9 ( + “ (ran 𝐹 × ran 𝐺)) = ran ( + ↾ (ran 𝐹 × ran 𝐺))
4442, 43eqtr4i 2767 . . . . . . . 8 ran 𝑃 = ( + “ (ran 𝐹 × ran 𝐺))
4540, 41, 443eltr4g 2855 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ ran 𝑃)
4625ffnd 6669 . . . . . . . 8 (𝜑𝐹 Fn ℝ)
47 dffn3 6681 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
4846, 47sylib 217 . . . . . . 7 (𝜑𝐹:ℝ⟶ran 𝐹)
4930ffnd 6669 . . . . . . . 8 (𝜑𝐺 Fn ℝ)
50 dffn3 6681 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
5149, 50sylib 217 . . . . . . 7 (𝜑𝐺:ℝ⟶ran 𝐺)
52 reex 11142 . . . . . . . 8 ℝ ∈ V
5352a1i 11 . . . . . . 7 (𝜑 → ℝ ∈ V)
54 inidm 4178 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5545, 48, 51, 53, 53, 54off 7635 . . . . . 6 (𝜑 → (𝐹f + 𝐺):ℝ⟶ran 𝑃)
5655frnd 6676 . . . . 5 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran 𝑃)
5756ssdifd 4100 . . . 4 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}))
5826sselda 3944 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
5931sselda 3944 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6058, 59anim12dan 619 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
61 readdcl 11134 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 + 𝑧) ∈ ℝ)
6260, 61syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺)) → (𝑦 + 𝑧) ∈ ℝ)
6362ralrimivva 3197 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
64 funimassov 7531 . . . . . . . 8 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6523, 36, 64sylancr 587 . . . . . . 7 (𝜑 → (( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6663, 65mpbird 256 . . . . . 6 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) ⊆ ℝ)
6744, 66eqsstrid 3992 . . . . 5 (𝜑 → ran 𝑃 ⊆ ℝ)
6867ssdifd 4100 . . . 4 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))
69 itg1val2 25048 . . . 4 (((𝐹f + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 ∖ {0}) ∈ Fin ∧ (ran (𝐹f + 𝐺) ∖ {0}) ⊆ (ran 𝑃 ∖ {0}) ∧ (ran 𝑃 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘(𝐹f + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹f + 𝐺) “ {𝑤}))))
703, 21, 57, 68, 69syl13anc 1372 . . 3 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹f + 𝐺) “ {𝑤}))))
7130adantr 481 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝐺:ℝ⟶ℝ)
7211adantr 481 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ran 𝐺 ∈ Fin)
73 inss2 4189 . . . . . . . . 9 ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
7473a1i 11 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
75 i1fima 25042 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
761, 75syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 “ {(𝑤𝑧)}) ∈ dom vol)
77 i1fima 25042 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
782, 77syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
79 inmbl 24906 . . . . . . . . . 10 (((𝐹 “ {(𝑤𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8076, 78, 79syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8180ad2antrr 724 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8219, 67sstrid 3955 . . . . . . . . . . . . 13 (𝜑 → (ran 𝑃 ∖ {0}) ⊆ ℝ)
8382sselda 3944 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℝ)
8483adantr 481 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℝ)
8559adantlr 713 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
8684, 85resubcld 11583 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤𝑧) ∈ ℝ)
8784recnd 11183 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ∈ ℂ)
8885recnd 11183 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
8987, 88npcand 11516 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) = 𝑤)
90 eldifsni 4750 . . . . . . . . . . . . 13 (𝑤 ∈ (ran 𝑃 ∖ {0}) → 𝑤 ≠ 0)
9190ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑤 ≠ 0)
9289, 91eqnetrd 3011 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧) + 𝑧) ≠ 0)
93 oveq12 7366 . . . . . . . . . . . . 13 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = (0 + 0))
94 00id 11330 . . . . . . . . . . . . 13 (0 + 0) = 0
9593, 94eqtrdi 2792 . . . . . . . . . . . 12 (((𝑤𝑧) = 0 ∧ 𝑧 = 0) → ((𝑤𝑧) + 𝑧) = 0)
9695necon3ai 2968 . . . . . . . . . . 11 (((𝑤𝑧) + 𝑧) ≠ 0 → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
9792, 96syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0))
98 itg1add.3 . . . . . . . . . . 11 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
991, 2, 98itg1addlem3 25062 . . . . . . . . . 10 ((((𝑤𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ ((𝑤𝑧) = 0 ∧ 𝑧 = 0)) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
10086, 85, 97, 99syl21anc 836 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) = (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
1011, 2, 98itg1addlem2 25061 . . . . . . . . . . 11 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
102101ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
103102, 86, 85fovcdmd 7526 . . . . . . . . 9 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
104100, 103eqeltrrd 2839 . . . . . . . 8 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10571, 72, 74, 81, 104itg1addlem1 25056 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
10683recnd 11183 . . . . . . . . 9 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → 𝑤 ∈ ℂ)
1071, 2i1faddlem 25057 . . . . . . . . 9 ((𝜑𝑤 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
108106, 107syldan 591 . . . . . . . 8 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → ((𝐹f + 𝐺) “ {𝑤}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧})))
109108fveq2d 6846 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑤})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
110100sumeq2dv 15588 . . . . . . 7 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {(𝑤𝑧)}) ∩ (𝐺 “ {𝑧}))))
111105, 109, 1103eqtr4d 2786 . . . . . 6 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑤})) = Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧))
112111oveq2d 7373 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹f + 𝐺) “ {𝑤}))) = (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)))
113103recnd 11183 . . . . . 6 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
11472, 106, 113fsummulc2 15669 . . . . 5 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · Σ𝑧 ∈ ran 𝐺((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
115112, 114eqtrd 2776 . . . 4 ((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · (vol‘((𝐹f + 𝐺) “ {𝑤}))) = Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
116115sumeq2dv 15588 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · (vol‘((𝐹f + 𝐺) “ {𝑤}))) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)))
11787, 113mulcld 11175 . . . . 5 (((𝜑𝑤 ∈ (ran 𝑃 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
118117anasss 467 . . . 4 ((𝜑 ∧ (𝑤 ∈ (ran 𝑃 ∖ {0}) ∧ 𝑧 ∈ ran 𝐺)) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
11921, 11, 118fsumcom 15660 . . 3 (𝜑 → Σ𝑤 ∈ (ran 𝑃 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
12070, 116, 1193eqtrd 2780 . 2 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
121 oveq1 7364 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦 + 𝑧) = ((𝑤𝑧) + 𝑧))
122 oveq1 7364 . . . . . . 7 (𝑦 = (𝑤𝑧) → (𝑦𝐼𝑧) = ((𝑤𝑧)𝐼𝑧))
123121, 122oveq12d 7375 . . . . . 6 (𝑦 = (𝑤𝑧) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
12418adantr 481 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ∈ Fin)
12567adantr 481 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝑃 ⊆ ℝ)
126125sselda 3944 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℝ)
12759adantr 481 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑧 ∈ ℝ)
128126, 127resubcld 11583 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → (𝑣𝑧) ∈ ℝ)
129128ex 413 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 → (𝑣𝑧) ∈ ℝ))
130126recnd 11183 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) → 𝑣 ∈ ℂ)
131130adantrr 715 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑣 ∈ ℂ)
13267sselda 3944 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ran 𝑃) → 𝑦 ∈ ℝ)
133132ad2ant2rl 747 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℝ)
134133recnd 11183 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑦 ∈ ℂ)
13559recnd 11183 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
136135adantr 481 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → 𝑧 ∈ ℂ)
137131, 134, 136subcan2ad 11557 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃)) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦))
138137ex 413 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → ((𝑣 ∈ ran 𝑃𝑦 ∈ ran 𝑃) → ((𝑣𝑧) = (𝑦𝑧) ↔ 𝑣 = 𝑦)))
139129, 138dom2lem 8932 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ)
140 f1f1orn 6795 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
141139, 140syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
142 oveq1 7364 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑧) = (𝑤𝑧))
143 eqid 2736 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))
144 ovex 7390 . . . . . . . 8 (𝑤𝑧) ∈ V
145142, 143, 144fvmpt 6948 . . . . . . 7 (𝑤 ∈ ran 𝑃 → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
146145adantl 482 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))‘𝑤) = (𝑤𝑧))
147 f1f 6738 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1→ℝ → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ)
148 frn 6675 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃⟶ℝ → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
149139, 147, 1483syl 18 . . . . . . . . . 10 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ⊆ ℝ)
150149sselda 3944 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑦 ∈ ℝ)
15159adantr 481 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝑧 ∈ ℝ)
152150, 151readdcld 11184 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦 + 𝑧) ∈ ℝ)
153101ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → 𝐼:(ℝ × ℝ)⟶ℝ)
154153, 150, 151fovcdmd 7526 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → (𝑦𝐼𝑧) ∈ ℝ)
155152, 154remulcld 11185 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
156155recnd 11183 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
157123, 124, 141, 146, 156fsumf1o 15608 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)))
158125sselda 3944 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℝ)
159158recnd 11183 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑤 ∈ ℂ)
160135adantr 481 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → 𝑧 ∈ ℂ)
161159, 160npcand 11516 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → ((𝑤𝑧) + 𝑧) = 𝑤)
162161oveq1d 7372 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ ran 𝑃) → (((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = (𝑤 · ((𝑤𝑧)𝐼𝑧)))
163162sumeq2dv 15588 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ ran 𝑃(((𝑤𝑧) + 𝑧) · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
164157, 163eqtrd 2776 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
16536ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (ran 𝐹 × ran 𝐺) ⊆ dom + )
166 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
167 simplr 767 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐺)
168166, 167opelxpd 5671 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺))
169 funfvima2 7181 . . . . . . . . . . . 12 ((Fun + ∧ (ran 𝐹 × ran 𝐺) ⊆ dom + ) → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
17023, 169mpan 688 . . . . . . . . . . 11 ((ran 𝐹 × ran 𝐺) ⊆ dom + → (⟨𝑦, 𝑧⟩ ∈ (ran 𝐹 × ran 𝐺) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺))))
171165, 168, 170sylc 65 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ( + ‘⟨𝑦, 𝑧⟩) ∈ ( + “ (ran 𝐹 × ran 𝐺)))
172 df-ov 7360 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + ‘⟨𝑦, 𝑧⟩)
173171, 172, 443eltr4g 2855 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ran 𝑃)
17458adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
175174recnd 11183 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
176135adantr 481 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
177175, 176pncand 11513 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) − 𝑧) = 𝑦)
178177eqcomd 2742 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 = ((𝑦 + 𝑧) − 𝑧))
179 oveq1 7364 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) → (𝑣𝑧) = ((𝑦 + 𝑧) − 𝑧))
180179rspceeqv 3595 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃𝑦 = ((𝑦 + 𝑧) − 𝑧)) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
181173, 178, 180syl2anc 584 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
182181ralrimiva 3143 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
183 ssabral 4019 . . . . . . 7 (ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)} ↔ ∀𝑦 ∈ ran 𝐹𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧))
184182, 183sylibr 233 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)})
185143rnmpt 5910 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) = {𝑦 ∣ ∃𝑣 ∈ ran 𝑃 𝑦 = (𝑣𝑧)}
186184, 185sseqtrrdi 3995 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
18759adantr 481 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
188174, 187readdcld 11184 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 + 𝑧) ∈ ℝ)
189101ad2antrr 724 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
190189, 174, 187fovcdmd 7526 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
191188, 190remulcld 11185 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℝ)
192191recnd 11183 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
193149ssdifd 4100 . . . . . . 7 ((𝜑𝑧 ∈ ran 𝐺) → (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹) ⊆ (ℝ ∖ ran 𝐹))
194193sselda 3944 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → 𝑦 ∈ (ℝ ∖ ran 𝐹))
195 eldifi 4086 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ ∖ ran 𝐹) → 𝑦 ∈ ℝ)
196195ad2antrl 726 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑦 ∈ ℝ)
19759adantr 481 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → 𝑧 ∈ ℝ)
198 simprr 771 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
1991, 2, 98itg1addlem3 25062 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
200196, 197, 198, 199syl21anc 836 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
201 inss1 4188 . . . . . . . . . . . . . . 15 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
202 eldifn 4087 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ ∖ ran 𝐹) → ¬ 𝑦 ∈ ran 𝐹)
203202ad2antrl 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑦 ∈ ran 𝐹)
204 vex 3449 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
205204eliniseg 6046 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V → (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦))
206205elv 3451 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (𝐹 “ {𝑦}) ↔ 𝑣𝐹𝑦)
207 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
208204, 207brelrn 5897 . . . . . . . . . . . . . . . . . . 19 (𝑣𝐹𝑦𝑦 ∈ ran 𝐹)
209206, 208sylbi 216 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑦 ∈ ran 𝐹)
210203, 209nsyl 140 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ¬ 𝑣 ∈ (𝐹 “ {𝑦}))
211210pm2.21d 121 . . . . . . . . . . . . . . . 16 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑣 ∈ (𝐹 “ {𝑦}) → 𝑣 ∈ ∅))
212211ssrdv 3950 . . . . . . . . . . . . . . 15 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝐹 “ {𝑦}) ⊆ ∅)
213201, 212sstrid 3955 . . . . . . . . . . . . . 14 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅)
214 ss0 4358 . . . . . . . . . . . . . 14 (((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ ∅ → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
215213, 214syl 17 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ∅)
216215fveq2d 6846 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = (vol‘∅))
217 0mbl 24903 . . . . . . . . . . . . . 14 ∅ ∈ dom vol
218 mblvol 24894 . . . . . . . . . . . . . 14 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
219217, 218ax-mp 5 . . . . . . . . . . . . 13 (vol‘∅) = (vol*‘∅)
220 ovol0 24857 . . . . . . . . . . . . 13 (vol*‘∅) = 0
221219, 220eqtri 2764 . . . . . . . . . . . 12 (vol‘∅) = 0
222216, 221eqtrdi 2792 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = 0)
223200, 222eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦𝐼𝑧) = 0)
224223oveq2d 7373 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) · 0))
225196, 197readdcld 11184 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℝ)
226225recnd 11183 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → (𝑦 + 𝑧) ∈ ℂ)
227226mul01d 11354 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · 0) = 0)
228224, 227eqtrd 2776 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ ∖ ran 𝐹) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0))) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
229228expr 457 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → (¬ (𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0))
230 oveq12 7366 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = (0 + 0))
231230, 94eqtrdi 2792 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦 + 𝑧) = 0)
232 oveq12 7366 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = (0𝐼0))
233 0re 11157 . . . . . . . . . . 11 0 ∈ ℝ
234 iftrue 4492 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
235 c0ex 11149 . . . . . . . . . . . 12 0 ∈ V
236234, 98, 235ovmpoa 7510 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → (0𝐼0) = 0)
237233, 233, 236mp2an 690 . . . . . . . . . 10 (0𝐼0) = 0
238232, 237eqtrdi 2792 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) → (𝑦𝐼𝑧) = 0)
239231, 238oveq12d 7375 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (0 · 0))
240 0cn 11147 . . . . . . . . 9 0 ∈ ℂ
241240mul01i 11345 . . . . . . . 8 (0 · 0) = 0
242239, 241eqtrdi 2792 . . . . . . 7 ((𝑦 = 0 ∧ 𝑧 = 0) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
243229, 242pm2.61d2 181 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
244194, 243syldan 591 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∖ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = 0)
245 f1ofo 6791 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃1-1-onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
246141, 245syl 17 . . . . . 6 ((𝜑𝑧 ∈ ran 𝐺) → (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)))
247 fofi 9282 . . . . . 6 ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)):ran 𝑃onto→ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
248124, 246, 247syl2anc 584 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧)) ∈ Fin)
249186, 192, 244, 248fsumss 15610 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣𝑧))((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
25019a1i 11 . . . . 5 ((𝜑𝑧 ∈ ran 𝐺) → (ran 𝑃 ∖ {0}) ⊆ ran 𝑃)
251117an32s 650 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ {0})) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) ∈ ℂ)
252 dfin4 4227 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))
253 inss2 4189 . . . . . . . 8 (ran 𝑃 ∩ {0}) ⊆ {0}
254252, 253eqsstrri 3979 . . . . . . 7 (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) ⊆ {0}
255254sseli 3940 . . . . . 6 (𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0})) → 𝑤 ∈ {0})
256 elsni 4603 . . . . . . . . 9 (𝑤 ∈ {0} → 𝑤 = 0)
257256adantl 482 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 = 0)
258257oveq1d 7372 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = (0 · ((𝑤𝑧)𝐼𝑧)))
259101ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝐼:(ℝ × ℝ)⟶ℝ)
260257, 233eqeltrdi 2846 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑤 ∈ ℝ)
26159adantr 481 . . . . . . . . . . 11 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → 𝑧 ∈ ℝ)
262260, 261resubcld 11583 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤𝑧) ∈ ℝ)
263259, 262, 261fovcdmd 7526 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℝ)
264263recnd 11183 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → ((𝑤𝑧)𝐼𝑧) ∈ ℂ)
265264mul02d 11353 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (0 · ((𝑤𝑧)𝐼𝑧)) = 0)
266258, 265eqtrd 2776 . . . . . 6 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ {0}) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
267255, 266sylan2 593 . . . . 5 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑤 ∈ (ran 𝑃 ∖ (ran 𝑃 ∖ {0}))) → (𝑤 · ((𝑤𝑧)𝐼𝑧)) = 0)
268250, 251, 267, 124fsumss 15610 . . . 4 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)) = Σ𝑤 ∈ ran 𝑃(𝑤 · ((𝑤𝑧)𝐼𝑧)))
269164, 249, 2683eqtr4d 2786 . . 3 ((𝜑𝑧 ∈ ran 𝐺) → Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
270269sumeq2dv 15588 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑤 ∈ (ran 𝑃 ∖ {0})(𝑤 · ((𝑤𝑧)𝐼𝑧)))
271192anasss 467 . . 3 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) ∈ ℂ)
27211, 9, 271fsumcom 15660 . 2 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
273120, 270, 2723eqtr2d 2782 1 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  Fincfn 8883  cc 11049  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056  cmin 11385  Σcsu 15570  vol*covol 24826  volcvol 24827  1citg1 24979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984
This theorem is referenced by:  itg1addlem5  25065
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